Examples of continuous but not uniformly continuous functions on various domains

Asked May 12 '23 at 19:35

Active May 12 '23 at 19:40

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What would be some simple examples of functions that are continuous but not uniformly continuous on the following intervals?

This is what I came up with:

Are these correct? Could it be that the third case is impossible? If so, why?

Thanks to anyone for the help!!

edited May 12 '23 at 19:40

Kenny Wong

asked May 12 '23 at 19:35

ugjumb

1

In the first example, what is $f(0)$? – Another User May 12 '23 at 19:37
Did you try $\sqrt x$? – Somos May 12 '23 at 19:44
Or even $f(x) = x^2$? – Kenny Wong May 12 '23 at 19:46
1

For the 3rd case, are you aware that all continuous functions on compact spaces are uniformly continuous? – Kenny Wong May 12 '23 at 19:46
Case 3: https://en.wikipedia.org/wiki/Heine%E2%80%93Cantor_theorem – Robert Z May 12 '23 at 19:46
Replace the first example with that one for instance. Your second example is fine, and so is your intuition about the 3rd question. – Anne Bauval May 12 '23 at 20:33
Would it matter if for the first one I take f(x)=x^2 or f(x)=x^n? – ugjumb May 13 '23 at 08:31
@AnneBauval And for the third to summarize: it is impossible because any function would be compact which is a finiteness property. So for uniformly continuous functions one particular delta works for every x and similarly compact functions have finitely many delta, which are enough to describe the whole space? – ugjumb May 13 '23 at 08:49
I don't know what a compact function is, but you seem to have the right ideas to prove the Heine-Cantor theorem (a proof can be found in every textbook on Topology, and in several posts on this site). – Anne Bauval May 13 '23 at 10:23
"Would it matter if for the first one I take f(x)=x^2 or f(x)=x^n? ": please look at the answer in the first link provided just before that comment of yours. – Anne Bauval May 13 '23 at 15:22

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